Chi-Square Calculation and the Meaning of p = 0.05
Mr Osei's SHS 2 class has analysed a maize genetics dataset: 1,100 offspring from a monohybrid cross, with 787 tall and 313 dwarf plants. The expected ratio is 3:1. In their notebooks, three students report different chi-square values: Akosua writes χ² = 0.27, Kwame writes χ² = 5.40, and Esi writes χ² = 7.00. Mr Osei knows that the critical value at p = 0.05 for one degree of freedom is 3.84. He wants to use the disagreement to teach both the calculation and the meaning of statistical significance, but he is conscious that many students are weak at converting fractions and at using squared numbers. The total of 1,100 is not evenly divisible by 4, which prompts several students to round the expected counts to 800 and 300.
1. Calculate the expected counts and the correct chi-square value, and identify which student is correct.
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Expected counts come from the 3:1 ratio applied to the total, not from convenient rounding. Total = 1,100, so the expected tall count is 3/4 × 1,100 = 825 and the expected dwarf count is 1/4 × 1,100 = 275. These sum to 1,100.
The chi-square statistic is the sum of (observed − expected)² divided by expected:
- Tall: (787 − 825)² / 825 = (−38)² / 825 = 1,444 / 825 = 1.750
- Dwarf: (313 − 275)² / 275 = (38)² / 275 = 1,444 / 275 = 5.251
- Total χ² = 7.001, approximately 7.00.
Esi is correct. Kwame's 5.40 and Akosua's 0.27 are not consistent with the full calculation. Degrees of freedom = number of categories − 1 = 2 − 1 = 1. The critical value at p = 0.05 is 3.84, and 7.00 > 3.84, so the null hypothesis of a 3:1 ratio is rejected at the 5% significance level. Rounding expected counts to 800 and 300 is not acceptable because the expected counts must preserve the total and the exact 3:1 proportion.
2. Explain what p = 0.05 means in this context and evaluate the statement 'χ² = 7.00 proves the plants did not follow a 3:1 ratio'.
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p = 0.05 is a decision threshold, not the probability that the 3:1 hypothesis is true or false. It means that if the null hypothesis (a true 3:1 ratio) were correct, there would be only a 5% chance of obtaining a deviation from the expected counts as large as, or larger than, the one observed. Because the calculated χ² of 7.00 exceeds the critical value of 3.84, the result falls in the rejection region, and we reject the null hypothesis at the 5% level.
The statement 'proves' is too strong. A significance test does not prove a biological hypothesis; it shows that the observed data are unlikely under the stated null model. The result could still arise from chance (5% of the time when the null is true), from sampling error, from non-random survival, or from a violation of assumptions such as independent offspring. The correct conclusion is: 'The data are inconsistent with a 3:1 ratio at the 5% significance level.' A biologist would then look for a biological explanation rather than treating the calculation as proof.
3. Suggest two numeracy supports Mr Osei could use to reduce calculation errors in future chi-square work.
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First, he should provide a structured calculation table with columns for observed, expected, observed − expected, (observed − expected)², and (observed − expected)²/expected. Students complete the table row by row, which prevents them from skipping the squaring step or dividing by the wrong number. The table also makes the expected-count rule visible: expected counts must sum to the same total as observed counts.
Second, he should teach a quick reasonableness check before any calculation. Students should estimate whether the observed counts are close to 3:1 by comparing 787:313 with 3:1; this helps them predict that χ² will be moderate rather than tiny. He can also require students to check that their final χ² cannot be negative and that a value such as 0.27 would imply an almost perfect fit, which is not what the raw counts suggest. These checks target the two common errors: rounding expected counts to 800 and 300, and mishandling squares or denominators.